Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,25)(21,24)(22,36)(23,28)(26,31)(27,37)(30,43)(32,40)(33,38)(34,47)(35,39)(41,44)(42,46)(45,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,21,23,31,37,35)(18,24,27,29,28,39,36,26)(20,30,34,33,32,44,46,45)(25,38,42,43,40,48,47,41);(2,5)(6,7)(17,20,23,32)(18,25,28,40)(19,30,31,44)(21,33,35,45)(22,34,37,46)(24,38,39,48)(26,41,29,43)(27,42,36,47);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(20,32)(25,40)(30,44)(33,45)(34,46)(38,48)(41,43)(42,47);(17,21,22,31,23,35,37,19)(18,26,27,24,28,29,36,39)(20,33,34,44,32,45,46,30)(25,41,42,38,40,43,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7498224);